In this paper, we propose a novel object matching method to match an object to its instance in an input scene image, where both the object template and the input scene image are represented by groups of feature points. We relax each template point's discrete feature cost function to create a convex function that can be optimized efficiently. Such continuous and convex functions with different regularization terms are able to create different convex optimization models handling objects undergoing (i) global transformation, (ii) locally affine transformation, and (iii) articulated transformation. These models can better constrain each template point's transformation and therefore generate more robust matching results. Unlike traditional object or feature matching methods with "hard" node-to-node results, our proposed method allows template points to be transformed to any location in the image plane. Such a property makes our method robust to feature point occlusion or mis-detection. Our extensive experiments demonstrate the robustness and flexibility of our method.